U Kv f9E@sVdZddlmZmZddlZddlmZddlm Z GdddZ Gdd d eZ dS) z Which Archimedean is Best? Extreme Value copulas formulas are based on Genest 2009 References ---------- Genest, C., 2009. Rank-based inference for bivariate extreme-value copulas. The Annals of Statistics, 37(5), pp.2990-3022. )ABCabstractmethodN)stats)utilsc@sBeZdZdZdddZdddZdd d Zdd d Zdd dZdS)CopulaDistributionaMultivariate copula distribution Parameters ---------- copula : :class:`Copula` instance An instance of :class:`Copula`, e.g. :class:`GaussianCopula`, :class:`FrankCopula`, etc. marginals : list of distribution instances Marginal distributions. copargs : tuple Parameters for copula Notes ----- Status: experimental, argument handling may still change cCs ||_||_||_t||_dSN)copula marginalscop_argslenk_vars)selfr r r rrL/tmp/pip-unpacked-wheel-2v6byqio/statsmodels/distributions/copula/copulas.py__init__'szCopulaDistribution.__init__NcCs|dkr|j}|dkr"dg|j}|jj|||d}t|jD]@\}}|jdd|dd|fdf|||dd|f<q>|S)a6Draw `n` in the half-open interval ``[0, 1)``. Sample the joint distribution. Parameters ---------- nobs : int, optional Number of samples to generate in the parameter space. Default is 1. cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute created when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. random_state : {None, int, numpy.random.Generator}, optional If `seed` is None then the legacy singleton NumPy generator. This will change after 0.13 to use a fresh NumPy ``Generator``, so you should explicitly pass a seeded ``Generator`` if you need reproducible results. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Returns ------- sample : array_like (n, d) Sample from the joint distribution. Notes ----- The random samples are generated by creating a sample with uniform margins from the copula, and using ``ppf`` to convert uniform margins to the one specified by the marginal distribution. See Also -------- statsmodels.tools.rng_qrng.check_random_state Nr)nobsargs random_stateg?gA?)r r r rvs enumerater Zppf)rrr marg_argsrsampleidistrrrr0s-   zCopulaDistribution.rvscCst|}|dkr|j}|dkr0dg|jd}g}t|jD],}||j|j|d|ff||q>t |}|j dkr| }|j ||S)aCDF of copula distribution. Parameters ---------- y : array_like Values of random variable at which to evaluate cdf. If 2-dimensional, then components of multivariate random variable need to be in columns cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute created when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. Returns ------- cdf values Nr.r) npasarrayr shaperanger appendr cdf column_stackndimsqueezer )ryr rcdf_margrurrrr!js *  zCopulaDistribution.cdfcCst|j|||dS)aPDF of copula distribution. Parameters ---------- y : array_like Values of random variable at which to evaluate cdf. If 2-dimensional, then components of multivariate random variable need to be in columns cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute created when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. Returns ------- pdf values )r r)rexplogpdf)rr%r rrrrpdfszCopulaDistribution.pdfcCst|}|dkr|j}|dkr4tdg|jd}d}g}t|jD]R}||j|j|d|ff||7}| |j|j |d|ff||qFt |}|j dkr| }||j||7}|S)aLog-pdf of copula distribution. Parameters ---------- y : array_like Values of random variable at which to evaluate cdf. If 2-dimensional, then components of multivariate random variable need to be in columns cop_args : tuple Copula parameters. If None, then the copula parameters will be taken from the ``cop_args`` attribute creating when initiializing the instance. marg_args : list of tuples Parameters for the marginal distributions. It can be None if none of the marginal distributions have parameters, otherwise it needs to be a list of tuples with the same length has the number of marginal distributions. The list can contain empty tuples for marginal distributions that do not take parameter arguments. Returns ------- log-pdf values Nrrg.r)rrr tuplerrr r r)r r!r"r#r$r )rr%r rZlpdfr&rr'rrrr)s &*  zCopulaDistribution.logpdf)r)rNNN)NN)NN)NN) __name__ __module__ __qualname____doc__rrr!r*r)rrrrrs  : ( rc@sxeZdZdZdddZddd Zedd d Zd d d Zed!ddZ d"ddZ d#ddZ d$ddZ ddZ ddZdS)%Copulau_A generic Copula class meant for subclassing. Notes ----- A function :math:`\phi` on :math:`[0, \infty]` is the Laplace-Stieltjes transform of a distribution function if and only if :math:`\phi` is completely monotone and :math:`\phi(0) = 1` [2]_. The following algorithm for sampling a ``d``-dimensional exchangeable Archimedean copula with generator :math:`\phi` is due to Marshall, Olkin (1988) [1]_, where :math:`LS^{−1}(\phi)` denotes the inverse Laplace-Stieltjes transform of :math:`\phi`. From a mixture representation with respect to :math:`F`, the following algorithm may be derived for sampling Archimedean copulas, see [1]_. 1. Sample :math:`V \sim F = LS^{−1}(\phi)`. 2. Sample i.i.d. :math:`X_i \sim U[0,1], i \in \{1,...,d\}`. 3. Return:math:`(U_1,..., U_d)`, where :math:`U_i = \phi(−\log(X_i)/V), i \in \{1, ...,d\}`. Detailed properties of each copula can be found in [3]_. Instances of the class can access the attributes: ``rng`` for the random number generator (used for the ``seed``). **Subclassing** When subclassing `Copula` to create a new copula, ``__init__`` and ``random`` must be redefined. * ``__init__(theta)``: If the copula does not take advantage of a ``theta``, this parameter can be omitted. * ``random(n, random_state)``: draw ``n`` from the copula. * ``pdf(x)``: PDF from the copula. * ``cdf(x)``: CDF from the copula. References ---------- .. [1] Marshall AW, Olkin I. “Families of Multivariate Distributions”, Journal of the American Statistical Association, 83, 834–841, 1988. .. [2] Marius Hofert. "Sampling Archimedean copulas", Universität Ulm, 2008. .. rvs[3] Harry Joe. "Dependence Modeling with Copulas", Monographs on Statistics and Applied Probability 134, 2015. cCs ||_dSr)k_dim)rr2rrrr szCopula.__init__rrNcCstdS)aEDraw `n` in the half-open interval ``[0, 1)``. Marginals are uniformly distributed. Parameters ---------- nobs : int, optional Number of samples to generate from the copula. Default is 1. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. random_state : {None, int, numpy.random.Generator}, optional If `seed` is None then the legacy singleton NumPy generator. This will change after 0.13 to use a fresh NumPy ``Generator``, so you should explicitly pass a seeded ``Generator`` if you need reproducible results. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Returns ------- sample : array_like (nobs, d) Sample from the copula. See Also -------- statsmodels.tools.rng_qrng.check_random_state NNotImplementedError)rrrrrrrr sz Copula.rvscCsdS)a[Probability density function of copula. Parameters ---------- u : array_like, 2-D Points of random variables in unit hypercube at which method is evaluated. The second (or last) dimension should be the same as the dimension of the random variable, e.g. 2 for bivariate copula. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. Returns ------- pdf : ndarray, (nobs, k_dim) Copula pdf evaluated at points ``u``. Nrrr'rrrrr*.sz Copula.pdfcCst|j|f|S)aYLog of copula pdf, loglikelihood. Parameters ---------- u : array_like, 2-D Points of random variables in unit hypercube at which method is evaluated. The second (or last) dimension should be the same as the dimension of the random variable, e.g. 2 for bivariate copula. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. Returns ------- cdf : ndarray, (nobs, k_dim) Copula log-pdf evaluated at points ``u``. )rlogr*r5rrrr)Csz Copula.logpdfcCsdS)akCumulative distribution function evaluated at points u. Parameters ---------- u : array_like, 2-D Points of random variables in unit hypercube at which method is evaluated. The second (or last) dimension should be the same as the dimension of the random variable, e.g. 2 for bivariate copula. args : tuple Arguments for copula parameters. The number of arguments depends on the copula. Returns ------- cdf : ndarray, (nobs, k_dim) Copula cdf evaluated at points ``u``. Nrr5rrrr!Xsz Copula.cdfcCsv|jdkrtd|dkr(|j||d}t|\}}||dddf|dddf|d|d||fS) aSample the copula and plot. Parameters ---------- sample : array-like, optional The sample to plot. If not provided (the default), a sample is generated. nobs : int, optional Number of samples to generate from the copula. random_state : {None, int, numpy.random.Generator}, optional If `seed` is None then the legacy singleton NumPy generator. This will change after 0.13 to use a fresh NumPy ``Generator``, so you should explicitly pass a seeded ``Generator`` if you need reproducible results. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. sample : array_like (n, d) Sample from the copula. See Also -------- statsmodels.tools.rng_qrng.check_random_state r1z#Can only plot 2-dimensional Copula.N)rrrrr'v)r2 ValueErrorrr create_mpl_axZscatter set_xlabel set_ylabel)rrrraxfigrrr plot_scatterms# $  zCopula.plot_scatter c Cs@ddlm}|jdkr(ddl}|dd}d}tt|d||t|d||\}}t| | gj } | | j |j } t| d } t| d } t|\} }tj| | |d }| | g}|j||| |d |d|dd }|d|d|dd|dd|d|j||d}|d| | S)aPlot the PDF. Parameters ---------- ticks_nbr : int, optional Number of color isolines for the PDF. Default is 10. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. r)pyplotr1NzPlotting 2-dimensional Copula.dg-C6?r_)numT)Z antialiasedZvminZvmaxr'r8equal)Zticksp)Z matplotlibrAr2warningswarnrZmeshgridZlinspaceZvstackZravelTr*ZreshaperZ nanpercentilerr:Zcontourfr;r<Zset_xlimZset_ylimZ set_aspectZcolorbarZ set_labelZ tight_layout)rZ ticks_nbrr=ZpltrHZ n_samplesZepsuuvvZpointsdataZmin_Zmax_r>ZvticksZ range_cbarcsZcbarrrrplot_pdfs<            zCopula.plot_pdfcCs6|j||d}t|dddf|dddfdS)zKendall's tau based on simulated samples. Returns ------- tau : float Kendall's tau. )rNrr)rr kendalltau)rrrxrrr tau_simulateds zCopula.tau_simulatedcstt|jddkrBtdddfdddfd}n(|jfddtD}t|}||S)aCopula correlation parameter using Kendall's tau of sample data. Parameters ---------- data : array_like Sample data used to fit `theta` using Kendall's tau. Returns ------- corr_param : float Correlation parameter of the copula, ``theta`` in Archimedean and pearson correlation in elliptical. If k_dim > 2, then average tau is used. rr1Nrc s@g|]8}t|dD]$}td|fd|fdqqS)r.r)rrrQ).0rjkrRrr sz)Copula.fit_corr_param..) rrrrrQr2rZmean _arg_from_tau)rrMtauZtausrrVrfit_corr_params *  zCopula.fit_corr_paramcCstdS)a@Compute correlation parameter from tau. Parameters ---------- tau : float Kendall's tau. Returns ------- corr_param : float Correlation parameter of the copula, ``theta`` in Archimedean and pearson correlation in elliptical. Nr3)rrZrrrrYszCopula._arg_from_tau)r1)rrN)r)r)r)Nr7NN)r@N)rPN)r,r-r.r/rrrr*r)r!r?rOrSr[rYrrrrr0s0  !    0 5 r0) r/abcrrZnumpyrZscipyrZstatsmodels.graphicsrrr0rrrrs   E